Armodue harmony: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2010-09-09 08:01:14 UTC</tt>.<br>

: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2010-09-09 08:38:11 UTC</tt>.<br>

: The original revision id was <tt>~~161353785~~</tt>.<br>

: The original revision id was <tt>161360407</tt>.<br>

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==The intervals of 3 eka and 13 eka==

The interval of 3 eka corresponds to the "wholetone" of Armodue (it is slightly wider than the tempered wholetone). This interval is particularly pleasing to the ear because it is very close to the natural tone that is formed with the seventh and the eighth harmonic (the tempered wholetone, by comparison, sounds less natural to the ear because it is formed with higher harmonics: the ninth and tenth). If you build scales using successions of the "wholetone" of Armodue, or proceeding for jumps of 3 eka, you get particularly evocative sounds - of vague pentatonic flavor.

The complement of the interval of 3 eka is the interval of 13 eka, which has a huge importance in Armodue as it corresponds to the natural minor seventh - the interval given by the ratio of the fourth harmonic with the seventh harmonic. Who has delved deeper into harmony topics is aware of how much a dominant seventh chord (example: C-E-G-Bb) played in 12-tone equal temperament differs from the corresponding natural chord found by overlapping the fourth, the fifth, the sixth and the seventh harmonic. This is due mainly to the non-negligible difference in pitch of the tempered minor seventh and the seventh harmonic (in this example the Bb). In Armodue, the minor seventh is returned to its natural pitch, the interval of 13 eka is perceived as very natural and euphonious.

From a philosophical point of view, with the two discussed intervals Armodue performs squaring the circle; within a "square", rigidly geometric structure (the division into sixteen notes), it places two "round", vague and exotic intervals as the natural wholetone and the natural minor seventh (in 3 and 13 eka). This latter consideration may prove as a very interesting opportunity for a composer.

The intervals of 3 and 13 eka are among the most suggestive intervals in Armodue and should be classified as sweet dissonances, as the tempered major second and minor seventh.

==The intervals of 4 eka and 12 eka==

XXX

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XXX</pre></div>

<h4>Original HTML content:</h4>

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Armodue armonia</title></head><body><div id="toc"><h1 class="nopad">Table of Contents</h1><div style="margin-left: 1em;"><a href="#Armodue: basic elements of harmony">Armodue: basic elements of harmony</a></div>

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Armodue armonia</title></head><body><div id="toc"><h1 class="nopad">Table of Contents</h1><div style="margin-left: 1em;"><a href="#Armodue: basic elements of harmony">Armodue: basic elements of harmony</a></div>

<div style="margin-left: 1em;"><a href="#Chapter 1: Two theses supporting the system">Chapter 1: Two theses supporting the system</a></div>

<div style="margin-left: 1em;"><a href="#Chapter 1: Two theses supporting the system">Chapter 1: Two theses supporting the system</a></div>

<div style="margin-left: 2em;"><a href="#Chapter 1: Two theses supporting the system-The supremacy of the fifth and and the seventh harmonic in Armodue">The supremacy of the fifth and and the seventh harmonic in Armodue</a></div>

<div style="margin-left: 2em;"><a href="#Chapter 1: Two theses supporting the system-The supremacy of the fifth and and the seventh harmonic in Armodue">The supremacy of the fifth and and the seventh harmonic in Armodue</a></div>

<div style="margin-left: 2em;"><a href="#Chapter 1: Two theses supporting the system-The triple mean of the double diagonal / side of the square">The triple mean of the double diagonal / side of the square</a></div>

<div style="margin-left: 2em;"><a href="#Chapter 1: Two theses supporting the system-The triple mean of the double diagonal / side of the square">The triple mean of the double diagonal / side of the square</a></div>

<div style="margin-left: 1em;"><a href="#Chapter 2: The interval table">Chapter 2: The interval table</a></div>

<div style="margin-left: 1em;"><a href="#Chapter 2: The interval table">Chapter 2: The interval table</a></div>

<div style="margin-left: 2em;"><a href="#Chapter 2: The interval table-Qualitative categories of intervals">Qualitative categories of intervals</a></div>

<div style="margin-left: 2em;"><a href="#Chapter 2: The interval table-Qualitative categories of intervals">Qualitative categories of intervals</a></div>

<div style="margin-left: 2em;"><a href="#Chapter 2: The interval table-The intervals of 1 eka and 15 eka">The intervals of 1 eka and 15 eka</a></div>

<div style="margin-left: 2em;"><a href="#Chapter 2: The interval table-The intervals of 1 eka and 15 eka">The intervals of 1 eka and 15 eka</a></div>

<div style="margin-left: 2em;"><a href="#Chapter 2: The interval table-The intervals of 2 eka and 14 eka">The intervals of 2 eka and 14 eka</a></div>

<div style="margin-left: 2em;"><a href="#Chapter 2: The interval table-The intervals of 2 eka and 14 eka">The intervals of 2 eka and 14 eka</a></div>

<div style="margin-left: 2em;"><a href="#Chapter 2: The interval table-The intervals of 3 eka and 13 eka">The intervals of 3 eka and 13 eka</a></div>

<div style="margin-left: 2em;"><a href="#Chapter 2: The interval table-The intervals of 3 eka and 13 eka">The intervals of 3 eka and 13 eka</a></div>

<div style="margin-left: ~~1em~~;"><a href="#Chapter 3: ~~Creating scales with Armodue: modal systems~~">~~Chapter 3: Creating scales with Armodue: modal systems~~</a>&lt~~;/div>~~

<div style="margin-left: 2em;"><a href="#Chapter 2: The interval table-The intervals of 4 eka and 12 eka">The intervals of 4 eka and 12 eka</a></div>

<div style="margin-left: 1em;"><a href="#Chapter 4: &quot;Geometric&quot; harmonic constructions with Armodue">Chapter 4: &quot;Geometric&quot; harmonic constructions with Armodue</a></div>

<div style="margin-left: 1em;"><a href="#Chapter 3: Creating scales with Armodue: modal systems">Chapter 3: Creating scales with Armodue: modal systems</a></div>

<div style="margin-left: 1em;"><a href="#Chapter 5: &quot;elastic&quot; chords">Chapter 5: &quot;elastic&quot; chords</a></div>

<div style="margin-left: 1em;"><a href="#Chapter 4: &quot;Geometric&quot; harmonic constructions with Armodue">Chapter 4: &quot;Geometric&quot; harmonic constructions with Armodue</a></div>

</div>

<div style="margin-left: 1em;"><a href="#Chapter 5: &quot;elastic&quot; chords">Chapter 5: &quot;elastic&quot; chords</a></div>

<h1 id="toc0"><a name="Armodue: basic elements of harmony"></a><strong>Armodue: basic elements of harmony</strong></h1>

</div>

<h1 id="toc0"><a name="Armodue: basic elements of harmony"></a><strong>Armodue: basic elements of harmony</strong></h1>

<br />

This is a translation of an article by Luca Attanasio. Original page in italian: <a class="wiki_link_ext" href="http://www.armodue.com/armonia.htm" rel="nofollow">http://www.armodue.com/armonia.htm</a><br />

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<br />

<h2 id="toc8"><a name="Chapter 2: The interval table-The intervals of 3 eka and 13 eka"></a>The intervals of 3 eka and 13 eka</h2>

XXX<br />

<br />

The interval of 3 eka corresponds to the &quot;wholetone&quot; of Armodue (it is slightly wider than the tempered wholetone). This interval is particularly pleasing to the ear because it is very close to the natural tone that is formed with the seventh and the eighth harmonic (the tempered wholetone, by comparison, sounds less natural to the ear because it is formed with higher harmonics: the ninth and tenth). If you build scales using successions of the &quot;wholetone&quot; of Armodue, or proceeding for jumps of 3 eka, you get particularly evocative sounds - of vague pentatonic flavor.<br />

<br />

The complement of the interval of 3 eka is the interval of 13 eka, which has a huge importance in Armodue as it corresponds to the natural minor seventh - the interval given by the ratio of the fourth harmonic with the seventh harmonic. Who has delved deeper into harmony topics is aware of how much a dominant seventh chord (example: C-E-G-Bb) played in 12-tone equal temperament differs from the corresponding natural chord found by overlapping the fourth, the fifth, the sixth and the seventh harmonic. This is due mainly to the non-negligible difference in pitch of the tempered minor seventh and the seventh harmonic (in this example the Bb). In Armodue, the minor seventh is returned to its natural pitch, the interval of 13 eka is perceived as very natural and euphonious.<br />

<br />

From a philosophical point of view, with the two discussed intervals Armodue performs squaring the circle; within a &quot;square&quot;, rigidly geometric structure (the division into sixteen notes), it places two &quot;round&quot;, vague and exotic intervals as the natural wholetone and the natural minor seventh (in 3 and 13 eka). This latter consideration may prove as a very interesting opportunity for a composer.<br />

<br />

The intervals of 3 and 13 eka are among the most suggestive intervals in Armodue and should be classified as sweet dissonances, as the tempered major second and minor seventh.<br />

<br />

<h2 id="toc9"><a name="Chapter 2: The interval table-The intervals of 4 eka and 12 eka"></a>The intervals of 4 eka and 12 eka</h2>

<br />

XXX<br />

<br />

<h1 id="~~toc9~~"><a name="Chapter 3: Creating scales with Armodue: modal systems"></a>Chapter 3: Creating scales with Armodue: modal systems</h1>

<h1 id="toc10"><a name="Chapter 3: Creating scales with Armodue: modal systems"></a>Chapter 3: Creating scales with Armodue: modal systems</h1>

<br />

XXX<br />

<br />

<h1 id="~~toc10~~"><a name="Chapter 4: &quot;Geometric&quot; harmonic constructions with Armodue"></a>Chapter 4: &quot;Geometric&quot; harmonic constructions with Armodue</h1>

<h1 id="toc11"><a name="Chapter 4: &quot;Geometric&quot; harmonic constructions with Armodue"></a>Chapter 4: &quot;Geometric&quot; harmonic constructions with Armodue</h1>

<br />

XXX<br />

<br />

<h1 id="~~toc11~~"><a name="Chapter 5: &quot;elastic&quot; chords"></a>Chapter 5: &quot;elastic&quot; chords</h1>

<h1 id="toc12"><a name="Chapter 5: &quot;elastic&quot; chords"></a>Chapter 5: &quot;elastic&quot; chords</h1>

<br />

XXX</body></html></pre></div>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2010-09-09 08:01:14 UTC</tt>.<br>
+: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2010-09-09 08:38:11 UTC</tt>.<br>
-: The original revision id was <tt>161353785</tt>.<br>
+: The original revision id was <tt>161360407</tt>.<br>
 : The revision comment was: <tt></tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
@@ Line 74: / Line 74: @@
 ==The intervals of 3 eka and 13 eka==
+The interval of 3 eka corresponds to the "wholetone" of Armodue (it is slightly wider than the tempered wholetone). This interval is particularly pleasing to the ear because it is very close to the natural tone that is formed with the seventh and the eighth harmonic (the tempered wholetone, by comparison, sounds less natural to the ear because it is formed with higher harmonics: the ninth and tenth). If you build scales using successions of the "wholetone" of Armodue, or proceeding for jumps of 3 eka, you get particularly evocative sounds - of vague pentatonic flavor.
+The complement of the interval of 3 eka is the interval of 13 eka, which has a huge importance in Armodue as it corresponds to the natural minor seventh - the interval given by the ratio of the fourth harmonic with the seventh harmonic. Who has delved deeper into harmony topics is aware of how much a dominant seventh chord (example: C-E-G-Bb) played in 12-tone equal temperament differs from the corresponding natural chord found by overlapping the fourth, the fifth, the sixth and the seventh harmonic. This is due mainly to the non-negligible difference in pitch of the tempered minor seventh and the seventh harmonic (in this example the Bb). In Armodue, the minor seventh is returned to its natural pitch, the interval of 13 eka is perceived as very natural and euphonious.
+From a philosophical point of view, with the two discussed intervals Armodue performs squaring the circle; within a "square", rigidly geometric structure (the division into sixteen notes), it places two "round", vague and exotic intervals as the natural wholetone and the natural minor seventh (in 3 and 13 eka). This latter consideration may prove as a very interesting opportunity for a composer.
+The intervals of 3 and 13 eka are among the most suggestive intervals in Armodue and should be classified as sweet dissonances, as the tempered major second and minor seventh.
+==The intervals of 4 eka and 12 eka==
 XXX
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 XXX</pre></div>
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   &lt;br /&gt;
 This is a translation of an article by Luca Attanasio. Original page in italian: &lt;a class="wiki_link_ext" href="http://www.armodue.com/armonia.htm" rel="nofollow"&gt;http://www.armodue.com/armonia.htm&lt;/a&gt;&lt;br /&gt;
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 &lt;br /&gt;
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-  XXX&lt;br /&gt;
+  &lt;br /&gt;
+The interval of 3 eka corresponds to the &amp;quot;wholetone&amp;quot; of Armodue (it is slightly wider than the tempered wholetone). This interval is particularly pleasing to the ear because it is very close to the natural tone that is formed with the seventh and the eighth harmonic (the tempered wholetone, by comparison, sounds less natural to the ear because it is formed with higher harmonics: the ninth and tenth). If you build scales using successions of the &amp;quot;wholetone&amp;quot; of Armodue, or proceeding for jumps of 3 eka, you get particularly evocative sounds - of vague pentatonic flavor.&lt;br /&gt;
+&lt;br /&gt;
+The complement of the interval of 3 eka is the interval of 13 eka, which has a huge importance in Armodue as it corresponds to the natural minor seventh - the interval given by the ratio of the fourth harmonic with the seventh harmonic. Who has delved deeper into harmony topics is aware of how much a dominant seventh chord (example: C-E-G-Bb) played in 12-tone equal temperament differs from the corresponding natural chord found by overlapping the fourth, the fifth, the sixth and the seventh harmonic. This is due mainly to the non-negligible difference in pitch of the tempered minor seventh and the seventh harmonic (in this example the Bb). In Armodue, the minor seventh is returned to its natural pitch, the interval of 13 eka is perceived as very natural and euphonious.&lt;br /&gt;
+&lt;br /&gt;
+From a philosophical point of view, with the two discussed intervals Armodue performs squaring the circle; within a &amp;quot;square&amp;quot;, rigidly geometric structure (the division into sixteen notes), it places two &amp;quot;round&amp;quot;, vague and exotic intervals as the natural wholetone and the natural minor seventh (in 3 and 13 eka). This latter consideration may prove as a very interesting opportunity for a composer.&lt;br /&gt;
+&lt;br /&gt;
+The intervals of 3 and 13 eka are among the most suggestive intervals in Armodue and should be classified as sweet dissonances, as the tempered major second and minor seventh.&lt;br /&gt;
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+&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="Chapter 2: The interval table-The intervals of 4 eka and 12 eka"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;The intervals of 4 eka and 12 eka&lt;/h2&gt;
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